Reactant fraction

Fig. 7.3 Kinetic plots for nC and 14C in the reaction of labeled methyl iodide with N,N-dimethyl- -toluidine in methanol at 30°C. (CR is the count rate (cpm) for the reactant fraction and CP the count rate for product fraction) (After Axelsson, B. S. et. al. J. Am. Chem. Soc. 109, 7233 (1987))...

Dialkyl disulphides, of which (S)-cystine above is an example, may be prepared from thiols by mild oxidation, usually with iodine in the presence of alkali. A convenient synthesis of unsymmetrical analogues results when a symmetrical disulphide is heated with thiol to establish an equilibrium mixture. By careful choice of the reactants, fractional distillation then removes the more volatile thiol leaving the mixed disulphide as a residue.240... 

We see from these results that the fractionation factors for the symmetrical transition states reflect the values for the reactant (and product). However, whereas the reactant fractionation factors span a range of 11% (from 0.97 to 1.08) the transition state factors are more widely dispersed covering a range of 18% (from 0.94 to 1.12). These results will be discussed further, but first we have to consider an argument advanced by Shiner (197 Id) in which the a-deuterium isotope effect is used to measure tjY. 

The usual interpretation of the parameter P, referred to here as the deposition modulus, is that it is the square of the ratio of the characteristic time for diffusion to the characteristic time for surface deposition. In this view it is equivalent to the square of the Thiele modulus commonly appearing in analyses of porous-bed catalysis. Another useful interpretation of this parameter is that it is the ratio of two rates - the rate of deposition on the preform fiber surfaces, Ss a, to the maximum rate of diffusive transport, pDDf/a. Thus when P is small, the actual rate of diffusive transport will be less than this maximum, and the mean gradient of the reactant fraction will be smaller than the maximum value off/a. Under any of these interpretations, small values of P are associated with high uniformity of both the reactant fraction and coating thickness. 

The derivative of the diffusivity on the right of Eq. 15 is the order of the diffusivity with respect to the reactant fraction. From Eq. 6 it is given by... 

However, under a rather broad range of conditions its value is near zero. For a simple binary mixture of the reactant and a single diluent gas, da/df is exactly zero because binary diffusivities of low-pressure gases do not depend on species concentrations. Further, effective binary diffusivity of the reactant and a gas mixture also shows no dependence on the reactant fraction provided that the composition of the mixture does not vary with the reactant fraction. This of course is the case whenever the reactant fraction is small. 

Under the simplifying assumption that the diffusivity is independent of the reactant fraction, the governing conservation Eq. 15 may be written as... 

A second useful normalization of the deposition rate is the simple modification of S given by S = Note that the normalized centerline deposition rate can be obtained from Eqs.22a, b simply by taking/ =f where/ is the normalized reactant fraction at z = 0. The pressure, temperature and all other variables in these expressions are uniform over the preform thickness. 

The deposition modulus at low temperatures is small, and the profile of the reactant concentration through the preform thickness is very uniform. In this case, the deposition rate at the center is nearly as large as that at the preform surface. With increasing temperature, the deposition modulus increases and the reactant concentration at the preform center falls. In this case, the centerline deposition rate becomes small relative to that at the surface. This behavior is illustrated in Figure 2. Here we see that the normalized centerline reactant fraction falls monotonically with increasing values of the deposition modulus. The centerline reactant fraction does not exhibit any sort of maximum, as is well known, and the deposition uniformity, U = /, falls smoothly as the deposition modulus is increased. 

Fig. 2 Distribution of normalized reactant fraction through the preform thickness. Reactant fractions depend only on the dimensionless reaction yield, /, and the deposition modulus, p.

Eor the special case of y 0, again corresponding to no net reaction yield or to a very small reactant fraction, Eq.l9 yields... 

Under the further restriction that corresponding to 7 0, the last term on the left of Eq. 36 may be neglected. This special case yields P 4.2656 and a corresponding centerline reactant fraction of 0.2495. The optimum temperature can then be obtained by using the optimum pressure,/ = 2, in Eq. 23. This gives... 

Form these two reactions the reactant fraction is given by f = I / G + 1) provided that the reactant flow rates are sufficiently large that product gases do not significantly accumulate inside the furnace. Under this restriction, the reactant fraction is f = 0.33 for a relative flow rate of G = 2. From the surface reaction we see that two moles of gas-phase product are produced for each mole of reactant, giving / = 1 and a normalized reactant yield of = /. = 0. 33. The parameters and b for the surface reaction probability are approximately 147 kJ/mol and 446, respectively. ... 

Optimum temperatures are obtained from the analytical expressions describing the optimum conditions, along with the derivative of the centerline reactant fraction with respect to the deposition modulus. Using an analytical solution to provide the derivative of the centerline reactant fraction, a closed-form implicit expression for the optimum temperature is obtained for the special case of no normalized reaction yield. For the more general case, this derivative is computed from numerical solutions to the equations governing transport and deposition. Optimum temperatures are presented graphically for a very wide range of the normalized preform thickness and normalized reaction yield. 

Crossley et al. (40) used a computer reduction technique for the DSC isothermal curve which was developed to replace the use of a planimeter. The data reduction was divided into two phases (1) mechanism-independent solutions for the reactant fraction, a, and various functions of a (where a is the reactant fraction remaining at time f) and (2) solutions for mechanism-dependent rate constants. For the first phase, the DATAR program was developed, which consisted of the following Ordinal points referred to a coarse data, and evenly spaced in time over the time span of the DSC curve, are read directly into the computer. Up to 1000 points may be read, but 40-50 are usually sufficient for acceptable accuracy. The resultant fraction remaining at time t is calculated by the equation I... 

From equation 9, will, in general, be a curved function of x. The extra information about the transition state is contained in this curvature. A simple general method of analysing such curvature, developed from that suggested by Albery and Davies [23], will now be presented. We here concentrate on fractionation in the transition state, but the approach applies also to equilibria and to reactant fractionation. 

If y < 0, either (a) there is reactant fractionation on individual sites which has not yet been accounted for, or (b) the reaction is taking place by parallel paths. 

Figure 3. Plot of equation 24 for ay against A each curve is labelled with its value of alb. All transition states have to be in the area between the a/b = 0 and a/b = 1 parabolas the broken lines allow for reactant fractionation.

We now turn to some reactions where the proton transfer is concerted with other changes taking place in the reaction co-ordinate, and in particular with the making or breaking of carbon oxygen bonds. Many of these reactions involve water as the reactant. The solvent isotope effect is simpler to use when water is either the acid or the base because we do not have to bother about reactant fractionation. 

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